Controlling Chaos through Compactification in Cosmological Models with a Collapsing Phase

TL;DR

This paper investigates how compactifying extra dimensions in cosmological models can delay or control chaotic behavior during contraction, potentially enabling smooth transitions to quantum regimes and informing cyclic universe theories.

Contribution

It identifies specific compactifications based on topological properties that suppress chaos in various gravity theories, including string and M-theory models.

Findings

Certain compactifications delay chaos until near the big crunch

Controlled chaos allows homogeneous, flat contraction until quantum effects dominate

Results applicable to ekpyrotic, cyclic, and pre-big bang cosmological models

Abstract

We consider the effect of compactification of extra dimensions on the onset of classical chaotic "Mixmaster" behavior during cosmic contraction. Assuming a universe that is well-approximated as a four-dimensional Friedmann-Robertson--Walker model (with negligible Kaluza-Klein excitations) when the contraction phase begins, we identify compactifications that allow a smooth contraction and delay the onset of chaos until arbitrarily close the big crunch. These compactifications are defined by the de Rham cohomology (Betti numbers) and Killing vectors of the compactification manifold. We find compactifications that control chaos in vacuum Einstein gravity, as well as in string theories with N = 1 supersymmetry and M-theory. In models where chaos is controlled in this way, the universe can remain homogeneous and flat until it enters the quantum gravity regime. At this point, the classical equations leading to chaotic behavior can no longer be trusted, and quantum effects may allow a smooth approach to the big crunch and transition into a subsequent expanding phase. Our results may be useful for constructing cosmological models with contracting phases, such as the ekpyrotic/cyclic and pre-big bang models.